Parameter tuning and chattering adjustment of Super-Twisting Sliding
Mode Control system for linear plants
Alessandro Pilloni, Alessandro Pisano and Elio Usai
Abstract— A simple procedure for tuning the parameters ofthe Super-Twisting (STW) second-order sliding mode control(2-SMC) algorithm, used for the feedback control of uncertainlinear plants, is presented. When the plant relative degree ishigher than one, it is known [10] that a self-sustained periodicoscillation takes place in the feedback system. The purpose ofthe present work is that of illustrating a systematic procedurefor control tuning based on Describing Function (DF) approachguaranteeing pre-specified frequency and magnitude of theresulting oscillation. The knowledge of the plant’s HarmonicResponse (magnitude and phase) at the desired chatteringfrequency is the only required prior information. By means of asimulation example, we show the effectiveness of the proposedprocedure.
I. INTRODUCTION
Sliding mode control (SMC) is a popular approach to
control system design under heavy uncertainty conditions
[25], and is accurate, robust and very simple to implement.
The main drawbacks of a classical first-order SMC (1-
SMC) are principally related to the so-called chattering effect
[7] [26], undesired steady-state vibrations of the system
variables.
A major cause of chattering has been identified as the
presence of unmodelled parasitic dynamics in the switching
devices [14]. To mitigate the chattering’s effects, many
solutions have been developed over the last three decades.
In particular three main approaches have been developed
• the use of a continuous approximation of the relay (e.g.
the saturation function [16];
• the use of an asymptotic state-observer to confine chat-
tering in the observer dynamics bypassing the plant [14];
• the use of higher-order sliding mode control algorithms
(HOSM) [3] [5] [21] [23] [4] [2].
In this paper we focus our attention on one of most
popular second order sliding mode algorithm (2-SM) known
as Super-Twisting algorithm [20]. Whenever applied to linear
plants with relative greater than one the STW controlled
system always exhibits chattering [10].
In this note, we first recall some known result (see
[10]) concerning the DF-based analysis of the dependence
of frequency and magnitude of chattering from the tuning
parameters of the STW controller. As the novelty, we propose
a procedure for selecting them, in order to assign prescribed
This work was not supported by any organizationA. Pilloni, A. Pisano and E. Usai are with the Department of Electrical
and Electronic Engineering (DIEE), University of Cagliari, Cagliari 09123,Italy
E-mail adresses: [email protected],[email protected], [email protected]
values to the frequency and amplitude of chattering. The
ability to affect the frequency of the residual steady state
oscillations may be useful, for example, to avoid resonant
behaviours of the plant.
In the literature there are two main approaches to chat-
tering analysis that provide an exact solution in terms of
magnitude and frequency of the periodic oscillation:
• time-domain analysis (i.e.: Poincare Maps [17] [19]);
• frequency domain techniques (i.e.: Tsypikin Locus [24]
and LPRS Method [8]).
All these approaches require lengthy computations. There-
fore, the application of approximate analysis methods has
been found useful whenever the plant under analysis present
low-pass filtering property [1]. Under this hypothesis, the
well-established DF method has been used to analyze the
characteristics of the periodic motions with 1-SMC [22] [27]
and 2-SMC algorithms, [10] [11] [12] [13], and the results
obtained via the use of exact techniques often feature a
satisfactory similarity to those obtained via the approximate
DF method [6].
To summarize, in this paper such representation is ex-
ploited for design purposes in the frequency domain in order
to provide effective tuning rules for chattering adjustment in
linear plants controlled by STW algorithm.
This paper is organized as follows: Section II presents the
Super-Twisting algorithm and recalls its DF-based analysis
[10]. Section III states the problem under investigation and
presents a simple graphical procedure for setting the param-
eters of the STW algorithm in order to assign prescribed
amplitude and frequency of the chattering motion. In Section
IV the proposed tuning procedure is verified by means of
computer simulations. Section V provides some concluding
remarks and hints for next research.
II. SUPER-TWISTING ALGORITHM AND ITS DF
ANALYSIS
We consider a linear SISO system, including actuator,
plant and measurement sensor, described by the following
state-space representation which comprises principal and
parasitic dynamics:
x(t) = Ax(t)+Bu(t), x ∈ Rn, u ∈ R
y(t) = Cx(t), y ∈ R(1)
where A, B, C are matrices of appropriate dimensions, x is
the state vector, u is the actuator’s input, y the plant’s output
and σ = r− y represent the error signal which is used as
sliding variable. We assume that the plant is asymptotically
12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012
978-1-4577-2067-3/12/$26.00 c©2011 IEEE 479
stable with low-pass filtering property. We shall use the plant
description in the form of transfer function as follows
W (s) =Y (s)
R(s)= C (sI −A)B (2)
The super-twisting control u(t) is given as follows [20]
u(t) = u1(t)+u2(t) (3)
u1 = −γsign(σ) , u1 (0) = 0 (4)
u2 = −λ | σ |ρ sign(σ) (5)
where λ , γ and ρ are positive design parameters, with 0.5≤ρ < 1. The control system under analysis can be represented
in the form of the block diagram in Fig. 1. The DF of the
nonlinear function (5) was derived in [10] [15] as follows:
N2 (ay) =2λa
ρ−1y
π
∫ π
0(sinψ)ρ+1dψ
=2λa
ρ−1y
√π
Γ( ρ
2+ 1
)
Γ(ρ
2+ 1.5
)
(6)
where ay is the oscillation amplitude of the sliding variable
σ (i.e., one of the two unknowns of the problem) and Γ is
the Gamma function. Let ρ = 0.5 then the DF formula (6)
specializes to
N2 (ay) =2λ
√ay
ay√
π
Γ(1.25)
Γ1.75≈
1.1128λ√ay
(7)
The DF of the nonlinear integral component (4) can be
written as follows:
N1 (ay,ω) =4γ
πay
1
jω(8)
which is the cascade connection of an ideal relay (with
the DF equal to 4γ/πay [1]), and an integrator with the
frequency response 1/ jω . With the account of both control
components, the DF of the Super-Twisting algorithm (3)-(5)
can be finally written as
N (ay,ω) = N1 (ay,ω)+N2 (ay)
=4γ
πay
1
jω+ 1.1128
λ√ay
(9)
Let us note that the DF of the super-twisting algorithm
depends on, both, the chattering amplitude ay and frequency
ω .
In general, the parameters of the limit cycle can be ap-
proximately found via the solution of the following complex
equation
1 +W ( jω) ·N (ay,ω) = 0 (10)
called the harmonic balance [1]. The harmonic balance
equation (10) can be rewritten as
W ( jω) = −N−1 (ay,ω) (11)
A periodic oscillation of frequency Ω and amplitude ay exists
when an intersection between the Nyquist Plot of the plant
Fig. 1. Block diagram of the system with the Super-Twisting Algorithm.
W ( jω) and the negative reciprocal of the DF, N−1(ay,ω),occurs at ω = Ω. Thus, the parameters of the limit cycle can
be found via solution of the harmonic balance equation (9)-
(10). The negative reciprocal of the DF (9) can be written in
explicit form as
−N−1(ay,ω) = −0.8986
√ay
λ+ j1.0282
γ
ωλ 2
1 + 1.3091ay
( γωλ
)2(12)
It is of interest to plot the negative reciprocal DF (12) in
the complex plane. It depends on the two variables ay and
ω ; which are both nonnegative by construction. It is clear
from (12) that with positive gains λ and γ the locus (12) is
entirely contained in the lower-left quadrant of the complex
plane when the variables ay and ω vary from zero to infinity.
Furthermore, we can see that, fixed ω = ω∗ = const,
every curves −N−1(ay,ω∗) admit an horizontal asymptote
at − j1.0282γ
ω∗λwhen ay → ∞. Also, it is easy to show that
limay→0
arg(−N−1(ay,ω∗) = −
π
2
In Fig. 2, the curves obtained for λ = 0.6 and γ = 0.8,
some values ω = ωi, and by letting ay to vary from 0 to ∞are displayed.
A. Existence of the Periodic Solution
The harmonic balance (10) can be also expressed as
N (ay,Ω) = −W−1( jΩ) (13)
which, considering (9), specializes to
4γ
πay
1
jΩ+ 1.1128
λ√ay
= −W−1( jΩ) (14)
Separating the complex equation (14) in its real and imagi-
nary part it yields
1.1128 λ√ay
= −ReW−1( jΩ)
4γπΩ
1ay
= − ImW−1( jΩ)
(15)
Deriving ay from the first of (15) and substituting this value
into the second of (15), we obtain the next equation
4γ
πΩ
1
ImW−1( jΩ)−
(
1.1128λ
ReW−1( jΩ)
)2
= 0 (16)
12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012
978-1-4577-2067-3/12/$26.00 c©2011 IEEE 480
which allows to compute the frequency Ω. Solution of
(16) cannot be derived in closed form, and a numerical,
or graphical approach is sought. Once obtained Ω , the
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−2
−1.5
−1
−0.5
0
0.5
Re
Im
ay
ωi
ω1
ω2
ω3
ωn
ωi>ω
i+1
Fig. 2. Plots of the negative reciprocal DF (12) for different values of ω .
amplitude of the periodic solution can be expressed as
ay =4γ
πΩ
1
ImW−1( jΩ)(17)
As noticed in [10], a point of intersection between the
Nyquist Plot of the plant and the negative reciprocal of
the DF always exists if the relative degree of the plant
transfer function is higher than one and it is located in the
third quadrant of the complex plane. From Fig. 2, it is also
apparent that the frequency of the periodic solution for the
Super-Twisting algorithm is always lower than the frequency
of the periodic motion for system controlled by conventional
relay.
The orbital asymptotic stability of the periodic solution
can be assessed using the well known Loeb Criterion [1]
[18], that is not mentioned here for the sake of brevity.
III. PROBLEM FORMULATION AND PROPOSED TUNING
PROCEDURE
A. Problem Formulation
Consider the feedback control system in Fig. 1, where
the plant is modeled by an unknown transfer function W (s)having relative degree greater than one. By means od the
previously outlined procedure, once determined the con-
troller gains λ and γ it can be approximately evaluated the
chattering parameters ay and Ω.
The task of this paper is that of giving a simple solution
to the “inverse” problem, reversing the objective from chat-
tering analysis (λ and γ fixed, ay and Ω to be computed) to
chattering adjustment (ay = ady and Ω = Ωd fixed and λ , γto be determined accordingly).
Hence, given the performance requirements in term of
desired frequency Ωd and maximal deviation ady of the chat-
tering motion, we shall define a graphical tuning procedure,
based on the DF method, for computing the parameters of
the algorithm (3).
To begin with, let us substitute (12) into (10) and rewrite
it as:
W ( jω) = −c1
a1.5y
λ
ay + c3
( γωλ
)2− j
c2ayγ
ωλ 2
ay + c3
( γωλ
)2(18)
with
c1 = 0.8986 , c2 = 1.0282 , c3 = 1.3091
Let
A(ω) =γ
ωλ, B(ω) =
γ
ω(19)
Multiplying both sides of (18) by B(ω), we derive
B(ω)W ( jω) = −c1a
1.5y A(ω)
ay + c3A2 (ω)
− jc2ayA
2 (ω)
ay + c3A2 (ω)
(20)
where once considered the design requirements ay = ady and
ω = Ωd , separating (20) in its real and imaginary part as
follows
B(
Ωd)
ReW(
jΩd)
= −c1a
dy
1.5A(Ωd)
ady+c3A2(Ωd)
B(
Ωd)
ImW(
jΩd)
= −c2a
dyA
2(Ωd)ady+c3A
2(Ωd)
(21)
we obtain a well-posed system of equations, where the only
unknown variables are A(Ωd) ≡ Ad and B(Ωd)≡ Bd . In fact
the harmonic response of the plant at ω = Ωd can be achieved
by a simple test on the plant W (s). Therefore, solving (21),
by the change of variables (19) we compute the values of λand γ that satisfy our requirements.
Remark 1: It is important to underline that it cannot be
achieved an arbitrary frequency for the chattering oscillation.
It is easy to conclude that Ωd must be chosen in the interval
Ω1 < Ωd < Ω2 (22)
with the lower and upper bounds such that
argW ( jΩ1) =π
2, argW ( jΩ2) = π (23)
In fact, the intersection between the Nyquist Plot of W ( jω)and the locus −N−1(ay,ω) always lies in the third quadrant
of the complex plane.
Direct solution of the nonlinear equation (20) can be
avoided by resorting to a graphical approach. It is convenient
to refer to the Figure 3, where each curve correspond to the
plot of the right-hand side of (20) in the complex plane, for
a specific constant values of ady and by letting A to vary from
0 to ∞.
Remark 2: It is important to point out the fact that the
right-hand side of (20) is independent of the plant transfer
function. Therefore the set of curves in Fig. 3 define an
abacus very useful to simplify the computation of the STW
control parameters.
12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012
978-1-4577-2067-3/12/$26.00 c©2011 IEEE 481
−0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
Re
Im
0.06
0.05
0.04
0.03
0.02
0.01
ay
0.07
0.08
deg−95deg−105deg−115
deg−125
deg−135
−145deg
deg−155
−165deg
deg−175
Fig. 3. The abacus for STW controller tuning.
B. How to use the Abacus
Now we illustrate step-by-step how to use the abacus in
Fig. 3 and how to compute the parameters of the controller.
1. Select ady and Ωd such that the constraint (22) holds
and determine the harmonic response of the plant through
the parameters∣
∣
∣W
(
jΩd)∣
∣
∣, arg
W(
jΩd)
(24)
2. Identify, among the several curves of the abacus,
(possibly by interpolation) that corresponding to the desired
chattering magnitude ady .
3. Draw in the abacus a segment starting from the origin
and forming an angel of arg
W ( jΩd)
with the positive real
axis (see Fig. 4). Let P the point of intersection between that
segment and the specific curve identified in the step 2.
4. Measure the length of the segment OP and afterwards
apply the next relations to derive A and B.
Bd = B(Ωd) = OP
|W( jΩd)|
Ad = A(Ωd) =c1
√ady
c2tan
arg
W(
jΩd)
(25)
5. According to (19) compute the values of λ and γ as
γ = ΩdBd
λ = γ
ΩdAd= Bd
Ad
(26)
IV. SIMULATION RESULTS
In order to outline the proposed method of tuning, we
consider the cascade connection of a linear plant P(s) and a
dynamic actuator A(s) both with a second-order dynamic.
P(s) = s+1s2+s+1
, A(s) = 10.0001s2+0.01s+1
W (s) = A(s) ·P(s)
(27)
Let us apply the described procedure presented in the previ-
ous section to shape the periodic solution parameters using
the STW controller (3) with ρ = 0.5.
Fig. 5. Step response of the plant W (s) controlled by a STW-2SM withλ ≈ 12.85, γ ≈ 90.5 and ρ = 0.5.
A. Let ady = 0.05 and Ωd = 80 rad/sec;
B. By frequency response test at ω = Ωd on the plant W (s)(27) we obtain:
|W ( jΩd)| ≈ 0.0143 , argW ( jΩd) ≈−155.78deg
C. Draw the segment OP in the abacus until it intersects
the curve associated to ady (see Fig. 4)
D. Calculate the distance OP in Fig. 4:
OP =√
(−0.0147)2 +(−0.0066)2 = 0.0161
E. By (25) we derive the next values for Ad and Bd
Ad = 0.0879 , Bd = 1.1303
and by (19) we compute the gains of the control
algorithm (3) that gives rise to the chattering motion
with the desired characteristics in the feedback control
system represented Fig. 1. One obtains
λ = 12.8565 , γ = 90.4238 . (28)
Figure 5 displays the unit step response of the closed loop
system with parameters (28). The zoomed sub-plot confirms
that the steady-state chattering motion fulfills the given spec-
ification of amplitude and frequency, therefore supporting the
presented analysis results.
V. CONCLUSIONS AND FUTURE WORK
A describing function approach for tuning a feedback
control system with a linear plant driven by STW algorithm
has been presented, which allows to shape the characteristics
of the chattering motion that occurs when the linear plant has
a relative degree greater than one.
A constructive procedure for determining in advance the
periodic solution parameters (frequency and amplitude) has
been developed and tested by means of computer simula-
tions.
Among some interesting directions for improving the
present result, the analysis, and shaping, of the transient
oscillations is of special interest. The mathematical treatment
presented in [9] and the “dynamic harmonic balance” concept
in particular, could be a possible starting point to this end.
12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012
978-1-4577-2067-3/12/$26.00 c©2011 IEEE 482
−10 −5−14.7
x 10−3
−0,012
−0,01
−0,008
−0,004
−0,002
0
−0.0066
Re
Im
−95deg−105deg−115deg
0.01
0.02
0.03
0.04
−175deg
deg−165
−155deg 0.05
−145deg
−125deg−135deg
P
O
Fig. 4. Example of abacus utilization for the system W(s) (27).
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